Problem 18 - College Algebra

Question

Solve each quadratic inequality in Exercises \(1-28\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ 3 x^{2}+16 x<-5 $$

Step-by-Step Solution

Answer

The detailed solution depends on the roots of the quadratic equation, that will be found using the quadratic formula. These roots will help in establishing and testing the intervals for the inequality and finally representing the solution in interval notation.

Step 1: Simplifying the inequality

Start by transforming the given inequality \(3x^{2} + 16x + 5 < 0\) into standard form by moving the value on the right to the left side of the inequality.

Step 2: Finding the roots

Solve the equation \(3x^{2} + 16x + 5 = 0\) using the quadratic formula to find the critical points \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here \(a = 3\), \(b = 16\), and \(c = 5\)

Step 3: Establishing the intervals

The roots will divide the number line into several intervals. These roots are the potential points where the inequality might shift its sign. Use these points to establish intervals on the number line.

Step 4: Testing the intervals

Pick a number from each interval and plug it into the original inequality. If the inequality holds, then that interval is part of the solution, otherwise, it's not.

Step 5: Representing solution in interval notation

Finally, represent the solution set of the quadratic inequality in interval notation. Levels of parentheses need to be used based on whether the interval is open or closed. Use '(' or ')' for open intervals and '[' or ']' for closed intervals.

Key Concepts

Interval Notation

Interval notation is a way to represent the set of solutions for an inequality or equation on a number line. It specifies the start and end points of the intervals, showing which numbers fulfill the inequality.

For example, considering the inequality we have, if our solution set includes all numbers greater than 2 but less than 5, we write this in interval notation as \( (2, 5) \). Here, the parentheses indicate open intervals, meaning 2 and 5 are not included.

If the boundaries are included, such as all numbers from 2 to 5, inclusive, we use brackets: \( [2, 5] \).

This notation helps to easily communicate a range of numbers without listing them individually. The choice of parentheses or brackets directly conveys whether the endpoints are part of the solution set or not.

Quadratic Formula

The quadratic formula is essential for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It allows us to find the roots, or solutions, of the quadratic equation which are the points where the function crosses the x-axis.

The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
where:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

In quadratic inequalities like \( 3x^2 + 16x + 5 < 0 \), we use this formula to determine the critical points, which help divide the number line into intervals. Solving this gives us two roots, representing the transition points where the inequality's truthfulness could change. These roots help in testing the intervals and determining which intervals are solutions to the inequality.

Solution Set

The solution set is the collection of all values that satisfy the inequality or equation. For quadratic inequalities, these values lie within specific intervals determined by the roots.

After calculating the roots and testing the intervals, the solution set is obtained. For the inequality \( 3x^2 + 16x + 5 < 0 \), the test involves selecting any number from each interval, substituting it into the inequality, and checking if the resultant expression is true.

If true, the whole interval belongs to the solution set. If not, the interval is excluded. The final result, expressed in interval notation, provides a clear, concise expression of all possible solutions. This solution set is then graphically represented on a number line for visual clarity.

Graphing Inequalities

Graphing inequalities on a number line helps us visually represent solution sets. This can make the process of understanding the solution clearer and helps in checking our work.

For quadratic inequalities, \( ax^2 + bx + c < 0 \), once we know the intervals of interest from our calculations (using the roots), we can draw them.

The number line is split based on the critical points. Open or closed circles represent whether the endpoints are included or not:

• An open circle, \( \circ \), is used for a point that is not in the solution set.
• A closed dot, \( \cdot \), indicates points that are included in the solution set.

By shading on this number line where the inequality holds true, it gives a complete picture of the solution at a glance. This graphical method is straightforward once you've identified the solution intervals and is an excellent way of verifying your interval notation solution.